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Existence of Positive Solutions to a Nonlocal Boundary Value Problem with Laplacian onTime Scales
Advances in Difference Equations volume 2010, Article number: 809497 (2010)
Abstract
The nonlocal boundary value problem, with Laplacian of the form , has been considered. Two existence criteria of at least one and three positive solutions are presented. The first one is based on the Four functionals fixed point theorem in the work of R. Avery et al. (2008), and the second one is based on the Five functionals fixed point theorem. Meanwhile an example is worked out to illustrate the main result.
1. Introduction
Due to the unification of the theory of differential and difference equations, there have been many investigations working on the existence of positive solutions to boundary value problems for dynamic equations on time scales. Also there is much attention paid to the study of multipoint boundary value problem with Laplacian; see [1–10].
For convenience, throughout this paper we denote as the Laplacian operator, that is, , . , where .
In [11], the author discussed the positive solutions of a point boundary value problem for a secondorder dynamic equation on a time scale
where , , and with . And he got the existence of at least two positive solutions of the above problem by means of a fixed point theorem in a cone.
Zhao and Ge [9] considered the following multipoint boundary value problem with onedimensional Laplacian:
where , , , , , . By using a fixed point theorem in a cone, they obtained the existence of at least one, two, or three positive solutions under some sufficient conditions.
Motivated by the above results, in this paper, we investigate the nonlocal boundary value problem with Laplacian
where and .
For convenience, we list the following hypotheses:

(H1)
, , , , ;

(H2)
and is not identically zero on any compact subinterval of ;

(H3)
and is not identically zero on any compact subinterval of , also it satisfies
(1.4)
By using the Four functionals fixed point theorem and Five functionals fixed point theorem, we obtain the existence criteria of at least one positive solution and three positive solutions for the BVP (1.3). As an application, an example is worked out finally. The remainder of this paper is organized as follows. Section 2 is devoted to some preliminary discussions. We give and prove our main results in Section 3.
2. Preliminaries
The basic definitions and notations on time scales can be found in [12, 13]. In the following, we will provide some background materials on the theory of cones in Banach spaces. For more details, please refer to [14, 15].
Definition 2.1.
Let be a Banach space. A nonempty, closed set is said to be a cone provided that the following hypotheses are satisfied:

(1)
if , , then

(2)
if , , then
Every cone induces a partial ordering "" on defined by if and only if
Definition 2.2.
A map is said to be a nonnegative continuous concave functional on a cone of a real Banach space if is continuous and for all and . Similarly, we say that the map is a nonnegative continuous convex functional on a cone of a real Banach space if is continuous and for all and .
Let and be nonnegative continuous concave functionals on , and let and be nonnegative continuous convex functionals on ; then for positive numbers and , define the sets
The following lemma can be found in [16].
Lemma 2.3 (four functionals fixed point theorem).
If P is a cone in a real Banach space E, and are nonnegative continuous concave functionals on , and are nonnegative continuous convex functionals on , and there exist nonnegative positive numbers and , such that
is a completely continuous operator, and is a bounded set. If
 (i)

(ii)
, for all with and

(iii)
, for all with

(iv)
, for all with and ,

(v)
, for all with
then has a fixed point in .
We are now in a position to present the Five functionals fixed point theorem (see [17]). Let be nonnegative continuous convex functionals on and nonnegative continuous concave functionals on . For nonnegative numbers and define the following convex sets:
Lemma 2.4 (five functionals fixed point theorem).
Let be a cone in a real Banach space . Suppose that there exist nonnegative numbers and , nonnegative continuous concave functionals and on , and nonnegative continuous convex functionals and on , with
Suppose that is completely continuous and there exist nonnegative numbers with such that

(i)
and for

(ii)
and for

(iii)
for with

(iv)
for with
then A has at least three fixed points such that
Consider the Banach space equipped with the norm . Suppose , with . For the sake of convenience, we take the notations
Define a cone
and an operator by
Lemma 2.5.
.
Proof.
For , ,
From the definition of , it is clear that
is continuous, and is the maximum value of on .
Let , then is continuous, is delta differentiable on , and is continuously differentiable. Moreover is monotonically increasing for and . Then by the chain rule [12, Theorem , page 31], we obtain
where is in the interval . So, . This completes the proof.
3. Main Results and an Example
Theorem 3.1.
Assume that (H1), (H2), and (H3) hold, if there exist constants , , , with , , and suppose that satisfies the following conditions:
(A1) for all
(A2) for all
then the BVP (1.3) has a fixed point such that
Define maps
and let , and be defined by (2.1).
In order to complete the proof of Theorem 3.1, we first need to prove the following lemma.
Lemma 3.2.
is bounded and is completely continuous.
Proof.
For all , , which means that is a bounded set.
According to Lemma 2.5, it is clear that .
In view of the continuity of , there exists a constant such that , for all , . Consider
which means that is uniformly bounded.
In addition, for all , we have
Applying the ArzelàAscoli theorem on time scales [18], one can show that is relatively compact.
Now we prove that is continuous. Let be a sequence in which converges to uniformly on . Because is relatively compact, the sequence admits a subsequence converging to uniformly on . In addition,
Observe that
Hence, by the Lebesgue's dominated convergence theorem on time scales [19], insert into the above equality and then let , we obtain
From the definition of , we know that on . This shows that each subsequence of uniformly converges to . Therefore the sequence uniformly converges to . This means that is continuous at . So, is continuous on since is arbitrary. Thus, is completely continuous. This completes the proof.
Proof of Theorem 3.1.
Let
Clearly, . By direct calculation,
So, which means that (i) in Lemma 2.3 is satisfied.
For all with and , we have , and for all with and , we obtain that . Hence, (ii) and (iv) in Lemma 2.3 are fulfilled.
For any with
and for all with ,
Thus (iii) and (v) in Lemma 2.3 hold true. So, by Lemma 2.3, the BVP (1.3) has a fixed point in . This completes the proof.
Theorem 3.3.
Assume that (H1), (H2), and (H3) hold. If there exist constants , with , , , , , further suppose that satisfies the following conditions:
(B1) for all
(B2) for all
then the BVP (1.3) has at least three positive solutions , and such that
Proof.
Define these maps
and let , , and be defined by (2.3). It is clear that
Using similar methods as those in Lemma 3.2, we obtain that is completely continuous. Thus, we only need to show that . Let , then
which implies that .
Let and
we can verify that . By calculation,
So, , , , which means that and are not empty.
For ,
and for ,
Thus (i) and (ii) in Lemma 2.4 hold.
On the other hand, for with , we have . And for with , we can obtain Thus, (iii) and (iv) in Lemma 2.4 hold.
So, by Lemma 2.4, we obtain that the BVP (1.3) has at least three positive solutions such that
This completes the proof.
Remark 3.4.
Let , , , we can find that the conditions of Theorem 3.1 are contained in Theorem 3.3.
Example 3.5.
Let , , consider the following eightpoint BVP:
where , , , , , , , , , , , , , for all , and
where is continuous, , and .
Set , , by calculation,
and let , , , , . Clearly, we can verify that the conditions in Theorem 3.3 are fulfilled. Thus, by Theorem 3.3, the BVP (3.21) has at least three positive solutions , and such that
Remark 3.6.
If we let , , , , we can also verify that the conditions in Theorem 3.1 are satisfied.
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Acknowledgments
This work is supported by the Natural Science Foundation of Ludong University (24200301, 24070301, 24070302), Program for Innovative Research Team in Ludong University, and a Project of Shandong Province Higher Educational Science and Technology Program.
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Keywords
 Banach Space
 Difference Equation
 Convergence Theorem
 Continuous Operator
 Chain Rule